The Kottwitz conjecture for parahoric test functions

Assume ${\bf G}_{\mathbb Q_{p^r}}$ is unramified and $K_p\subset{\bf G}(\mathbb Q_p)$ is parahoric. Let $d={\rm \dim}(Sh_{K_p})$, and let $z_{-\mu,j}$ be the Bernstein function associated to the conjugacy class $\{\mu\}$. Kottwitz conjecture. The test function $\phi_r$ in the semisimple Lefschetz formula may be taken to be

$$\phi_r=p^{rd/2}z_{-\mu,j}.$$

This is the unramified parahoric specialization of the test function conjecture and was formulated by Kottwitz; the source discusses proofs in several special cases.